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Proof Assistants: history, ideas and future
"... In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assista ..."
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In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assistants are used and how we envision their extended use in the future. While being an introduction into the world of proof assistants and the main issues behind them, this paper is also a position paper that pushes the further use of proof assistants. We believe that these systems will become the future of mathematics, where definitions, statements, computations and proofs are all available in a computerized form. An important application is and will be in computer supported modelling and verification of systems. But their is still along road ahead and we will indicate what we believe is needed for the further proliferation of proof assistants.
Teaching logic using a stateoftheart proof assistant
, 2007
"... This article describes the system ProofWeb that is currently being developed in Nijmegen and Amsterdam for teaching logic to undergraduate computer science students. This system is based on the higher order proof assistant Coq, and is made available to the students through an interactive web interfa ..."
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This article describes the system ProofWeb that is currently being developed in Nijmegen and Amsterdam for teaching logic to undergraduate computer science students. This system is based on the higher order proof assistant Coq, and is made available to the students through an interactive web interface. Part of this system will be a large database of logic problems. This database will also hold the solutions of the students. This means that the students do not need to install anything to be able to use the system (not even a browser plugin), and that the teachers will be able to centrally track progress of the students. The system makes the full power of Coq available to the students, but simultaneously presents the logic problems in a way that is customary in undergraduate logic courses. Both styles of presenting natural deduction proofs (Gentzen style ‘tree view ’ and Fitch style ‘box view’) are supported. Part of the system is a parser that indicates whether the students used the automation of Coq to solve their problems or that they solved it themselves using only the inference rules of the logic. For these inference rules dedicated tactics for Coq have been developed. The system has already been used in a type theory course, and is currently being further developed in the first year logic course of computer science in Nijmegen.
Flyspeck in a Semantic Wiki Collaborating on a Large Scale Formalization of the Kepler Conjecture
"... Abstract. Semantic wikis have been successfully applied to many problems in knowledge management and collaborative authoring. They are particularly appropriate for scientific and mathematical collaboration. In previous work we described an ontology for mathematical knowledge based on the semantic ma ..."
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Abstract. Semantic wikis have been successfully applied to many problems in knowledge management and collaborative authoring. They are particularly appropriate for scientific and mathematical collaboration. In previous work we described an ontology for mathematical knowledge based on the semantic markup language OMDoc and a semantic wiki using both. We are now evaluating these technologies in concrete application scenarios. In this paper we evaluate the applicability of our infrastructure to mathematical knowledge management by focusing on the Flyspeck project, a formalization of Thomas Hales ’ proof of the Kepler Conjecture. After describing the Flyspeck project and its requirements in detail, we evaluate the applicability of two wiki prototypes to Flyspeck, one based on Semantic MediaWiki and another on our mathematicsspecific semantic wiki SWiM. 1 Scientific Communication and the Flyspeck Project Scientific communication consists mainly of exchanging documents, and a great deal of scientific work consists of collaboratively authoring them. Common instances are writing down first hypotheses, commenting on results of experiments or project steps, and structuring, annotating, or reorganizing existing items of knowledge, as depicted in Buchberger’s figure on the right.
Easily Editing and Browsing Complex OpenMath Markup with SWiM
, 2008
"... We present how the mathematical semantic wiki SWiM has been enhanced towards support for editing content dictionaries (CDs) in the semantic markup language OpenMath. The ongoing revision of the OpenMath CDs for the 3rd version of the standard has motivated several enhancements to the SWiM user inter ..."
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We present how the mathematical semantic wiki SWiM has been enhanced towards support for editing content dictionaries (CDs) in the semantic markup language OpenMath. The ongoing revision of the OpenMath CDs for the 3rd version of the standard has motivated several enhancements to the SWiM user interface: a structural editor for CDs and their most common elements, particularly addressing notation definitions for symbols, the integration of a visual OpenMath formula editor, and a formbased editor for metadata. We show how OpenMath CDs imported from the filesystem or a Subversion repository are split into handy fragments that are convenient to edit and navigate, and reassembled on export. 1
Position paper: A real Semantic Web for mathematics deserves a real semantics
"... Abstract. Mathematical documents, and their instrumentation by computers, have rich structure at the layers of presentation, metadata and semantics, as objects in a system for formal mathematical logic. Semantic Web tools [2] support the first two of these, with little, if any, contribution to the t ..."
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Abstract. Mathematical documents, and their instrumentation by computers, have rich structure at the layers of presentation, metadata and semantics, as objects in a system for formal mathematical logic. Semantic Web tools [2] support the first two of these, with little, if any, contribution to the third, while Proof Assistants [17] instrument the third layer, typically with bespoke approaches to the first two. Our position is that a web of mathematical documents, definitions and proofs should be given a fullyfledged semantics in terms of the third layer. We propose a “MathWiki ” to harness Web 2.0 tools and techniques to the rich semantics furnished by contemporary Proof Assistants. 1 Background and state of the art We can identify four worlds of mathematical discourse available on the Web: – Traditional mathematical practice: a systematic body of knowledge, organised around documents written by experts, most often in L ATEX, to varying degrees of sophistication. The intended audience is an expert readership, and
FORMED 2008 Teaching logic using a stateoftheart proof assistant
"... This article describes the system ProofWeb developed for teaching logic to undergraduate computer science students. The system is based on the higher order proof assistant Coq, and is made available to the students through an interactive web interface. Part of this system is a large database of lo ..."
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This article describes the system ProofWeb developed for teaching logic to undergraduate computer science students. The system is based on the higher order proof assistant Coq, and is made available to the students through an interactive web interface. Part of this system is a large database of logic problems. This database will also hold the solutions of the students. The students do not need to install anything to be able to use the system (not even a browser plugin), and the teachers are able to centrally track progress of the students. The system makes the full power of Coq available to the students, but simultaneously presents the logic problems in a way that is customary in undergraduate logic courses. Both styles of presenting natural deduction proofs (Gentzenstyle ‘tree view ’ and Fitchstyle ‘box view’) are supported. Part of the system is a parser that indicates whether the students used the automation of Coq to solve their problems or that they solved it themselves using only the inference rules of the logic. For these inference rules dedicated tactics for Coq have been developed. The system
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"... Abstract To improve on existing models of interaction with a proof assistant (PA), in particular for storage and replay of proofs, we introduce three related concepts, those of: a proof movie, consisting of frames which record both user input and the corresponding PA response; a camera, which films ..."
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Abstract To improve on existing models of interaction with a proof assistant (PA), in particular for storage and replay of proofs, we introduce three related concepts, those of: a proof movie, consisting of frames which record both user input and the corresponding PA response; a camera, which films a user’s interactive session with a PA as a movie; and a proviola, which replays a movie framebyframe to a third party. In this paper we describe the movie data structure and we discuss a prototype implementation of the camera and proviola based on the ProofWeb system [7]. ProofWeb uncouples the interaction with a PA via a webinterface (the client) from the actual PA that resides on the server. Our camera films a movie by “listening ” to the ProofWeb communication. The first reason for developing movies is to uncouple the reviewing of a formal proof from the PA used to develop it: the movie concept enables users to discuss small code fragments without the need to install the PA or to load a whole library into it. Other advantages include the possibility to develop a separate commentary track to discuss or explain the PA interaction. We assert that a combined camera+proviola provides a generic layer between a client (user) and a server (PA). Finally we claim that movies are the right type of data to be stored in an encyclopedia of formalized mathematics, based on our experience in filming the Coq standard library. 1
Communicating Formal Proofs: The Case of Flyspeck
"... Abstract. We introduce a platform for presenting and crosslinking formal and informal proof developments together. The platform supports writing natural language ‘narratives ’ that include islands of formal text. The formal text contains hyperlinks and gives ondemand state information at every pro ..."
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Abstract. We introduce a platform for presenting and crosslinking formal and informal proof developments together. The platform supports writing natural language ‘narratives ’ that include islands of formal text. The formal text contains hyperlinks and gives ondemand state information at every proof step. We argue that such a system significantly lowers the threshold for understanding formal development and facilitates collaboration on informal and formal parts of large developments. As an example, we show the Flyspeck formal development (in HOL Light) and the Flyspeck informal mathematical text as a narrative linked to the formal development. To make this possible, we use the Agora system, a MathWiki platform developed at Nijmegen which has so far mainly been used with the Coq theorem prover: we show that the system itself is generic and easily adapted to the HOL Light case. 1